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ON SOME VERTEX ALGEBRAS RELATED TO \( {V}_{-1}\left(\mathfrak{sl}(n)\right) \) AND THEIR CHARACTERS

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We consider several vertex operator algebras and superalgebras closely related to \( {V}_{-1}\left(\mathfrak{sl}(n)\right) \), n ≥ 3 : (a) the parafermionic subalgebra K(\( \mathfrak{sl} \)(n); −1) for which we completely describe its inner structure, (b) the vacuum algebra Ω(V1(\( \mathfrak{sl} \)(n))), and (c) an infinite extension \( \mathcal{U} \) of V1(\( \mathfrak{sl} \)(n)) obtained from certain irreducible ordinary modules with integral conformal weights. It turns out that \( \mathcal{U} \) is isomorphic to the coset vertex algebra \( \mathfrak{psl} \)(n|n)1/\( \mathfrak{sl} \)(n)1, n ≥ 3. We show that V1(\( \mathfrak{sl} \)(n)) admits precisely n ordinary irreducible modules, up to isomorphism. This leads to the conjecture that \( \mathcal{U} \) is quasi-lisse.We present evidence in support of this conjecture: we prove that the (super)character of \( \mathcal{U} \) is quasimodular of weight one by virtue of being the constant term of a meromorphic Jacobi form of index zero. Explicit formulas and MLDE for characters and supercharacters are given for \( \mathfrak{g} \) = \( \mathfrak{sl} \)(3) and outlined for general n. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras \( \mathfrak{psl} \)(n|n)1, n ≥ 2. We finish with a theorem pertaining to characters of \( \mathfrak{psl} \)(n|n)1 and \( \mathcal{U} \)-modules.

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Correspondence to ANTUN MILAS.

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Dedicated to Mirko Primc on the occasion of his 70th birthday

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Dražen Adamović is supported by the Croatian Science Foundation under the project 2634 and by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004)

Antun Milas is partially supported by the NSF grant DMS 1601070.

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ADAMOVIĆ, D., MILAS, A. ON SOME VERTEX ALGEBRAS RELATED TO \( {V}_{-1}\left(\mathfrak{sl}(n)\right) \) AND THEIR CHARACTERS. Transformation Groups 26, 1–30 (2021). https://doi.org/10.1007/s00031-020-09617-w

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